triadic harmonic entropy

In light of some recent discussions with Pierre and Margo, I thought it
would be worthwhile going back to various ways of looking at triads, in
anticipation of looking at larger collections of notes.

As a complement to http://www.egroups.com/files/tuning/triads.jpg, which
featured in our discussions in March, I've created
http://www.egroups.com/files/tuning/enttriad.jpg. While the former was a
step toward defining triadic harmonic entropy, the latter just takes the
regular, dyadic harmonic entropies for the three intervals in the triad, and
adds them. This, of course, ignores any harmonic/subharmonic distinctions,
so the resulting diagram is symmetrical about a diagonal axis.

Looking at the diagram, the lowest basins (most pairwise concordant triads)
within the range depicted are the dark blue ones: root-position major
(4:5:6), root-position minor (10:12:15), first-inversion minor (12:15:20),
second-inversion major (3:4:5), and stacked-fourths (9:12:16) triads. In the
stacked-fourths triad, the concordance of the fourths is enough to
compensate for the discordance of the 9:16 minor seventh. The entire valley
of chords where the outer interval is a perfect fifth is bluish (the blue
stripe running from left to bottom), reflecting the great concordance of the
perfect fifth (3:2). Slightly higher basins (light blue) are the
first-inversion major (5:6:8) and second-inversion minor (15:20:24) triads,
reflecting the fact that the minor sixth is not a highly concordant interval
. . . . more later . . .

> [Paul Erlich, TD 744.10]
>
> In light of some recent discussions with Pierre and Margo, I
> thought it would be worthwhile going back to various ways of
> looking at triads, in anticipation of looking at larger
> collections of notes.
>
> As a complement to http://www.egroups.com/files/tuning/triads.jpg,
> which featured in our discussions in March, I've created
> http://www.egroups.com/files/tuning/enttriad.jpg. While the
> former was a step toward defining triadic harmonic entropy, the
> latter just takes the regular, dyadic harmonic entropies for the
> three intervals in the triad, and adds them. This, of course,
> ignores any harmonic/subharmonic distinctions, so the resulting
> diagram is symmetrical about a diagonal axis.

Wow, Paul, these graphs are fantastic !!!

I'm particularly intrigued by the first one, regarding
triadic harmonic entropy. I've just spent hours reading
the follow-ups to that post. Too bad I missed them at
the time! (You posted the Voronoi diagram just when I was
arriving in California at the end of my long road trip.)

I think you're really on to something here - something that
many of us have been searching for for quite some time,
and that it merits much more subsequent research!

If you put together a coherent discussion of this with both
text and graphics, I'll be happy to give it a home at the
Sonic Arts website. Feedback from others appreciated too.

-monz

Joseph L. Monzo San Diego monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
| 'I had broken thru the lattice barrier...' |
| -Erv Wilson |
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